Mathematics: Science of arbitrary structures.

This is a somewhat tendentious definition, in that there are quite a few other definitions of what mathematics is or is about that also seem somewhat adequate, and perhaps also somewhat inadequate. One merit of the given definition is that it makes clear that everything - that has or is some structure, but what hasn't? - is somehow also studied by mathematics, that may contain a lot of provable statements about those kinds of structures, or aspects of them.

Something to the effect of the above definition was proposed by a number of French mathematicians, who cooperated under the alias of "Nicholas Bourbaki". A readable and interesting book that expounds these ideas is Frederik Muller's "Structures for everyone".